If it is muttar to tear a single thread if one is not mechaven to tear it to a specific shiur, would it then be muttar to rip dental floss on Shabbos?
The reason it is permitted to rip the string that is tied around a box, when there is no other option is because ripping the string is mekalkel, ruining the string. Regarding the dental floss, ripping the amount of floss that is needed is making the floss easier to use. Therefore, it is considered a tikkun and not permitted. If you want to use dental floss, (and it will not make you teeth bleed), you can use it while it is still attached to the roll.
SS”K New Edition 14-39, ftnt. 196.